- 57 Views
- Uploaded on

Download Presentation
## PowerPoint Slideshow about ' MVS 250: V. Katch' - lane-cherry

**An Image/Link below is provided (as is) to download presentation**

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript

Paired Data

is there a relationship

if so, what is the equation

use the equation for prediction

OverviewCorrelation

exists between two variables when one of them is related to the other in some way

Definition 1. The sample of paired data (x,y) is a random sample.

2. The pairs of (x,y) data have a bivariate normal distribution.

AssumptionsScatterplot (or scatter diagram)

is a graph in which the paired (x,y) sample data are plotted with a horizontal x axis and a vertical y axis. Each individual (x,y) pair is plotted as a single point.

Definitionr =

(SDx) (SDy)

- Definition
- Linear Correlation Coefficient r

measures strength of the linear relationship between paired x and y values in a sample

Where xy/n is the mean of the cross products; (x/n) is the mean of the x variable; (y/n) is the mean of the y variable; SDx is the standard deviation of the x variable and SDy is the standard deviation of the x variable

n number of pairs of data presented

denotes the addition of the items indicated.

x/ndenotes the mean of all x values.

y/ndenotes the mean of all y values.

xy/ndenotes the mean of the cross products [x times y, summed; divided by n]

r linear correlation coefficient for a sample

linear correlation coefficient for a population

Notation for the Linear Correlation Coefficient Round to three decimal places

Use calculator or computer if possible

Rounding the Linear Correlation Coefficient r

1. -1 r 1

2. Value of r does not change if all values of either variable are converted to a different scale.

3. The r is not affected by the choice of x and y. Interchange x and y and the value of r will not change.

4. r measures strength of a linear relationship.

Properties of the Linear Correlation Coefficient rIf the absolute value of r exceeds the value in Sig. Table, conclude that there is a significant linear correlation.

Otherwise, there is not sufficient evidence to support the conclusion of significant linear correlation.

Remember to use n-2

Interpreting the Linear Correlation Coefficient 1. Causation: It is wrong to conclude that correlation implies causality.

2. Averages: Averages suppress individual variation and may inflate the correlation coefficient.

3. Linearity: There may be some relationship between x and y even when there is no significant linear correlation.

Common Errors Involving CorrelationCorrelation Calculations

Rank Order Correlation -RhoPearson’s - r

Rank Order Correlation, cont

Rho = 1- [6 (∑D2) / N (N2-1)]

Rho = 1- [6(58)/10(102-1)]

Rho = 1- [348 / 10 (100 -1)]

Rho = 1- [348 / 990]

Rho = 1- 0.352

Rho = 0.648

(∑D2 = 58)

N=10

r =

(SDx) (SDy)

Pearson’s rr = 32.86 - (5.5) (5.5)/(3.03) (3.03)

r = 35.86 - 30.25 / 9.09

r = 5.61 / 9.09

r = 0.6172

Pearson’s r

Excel Demonstration

x Plastic (lb)

0.27

2

1.41

3

2.19

3

2.83

6

2.19

4

1.81

2

0.85

1

3.05

5

y Household

Is there a significant linear correlation?

x Plastic (lb)

0.27

2

1.41

3

2.19

3

2.83

6

2.19

4

1.81

2

0.85

1

3.05

5

y Household

Is there a significant linear correlation?

x Plastic (lb)

0.27

2

1.41

3

2.19

3

2.83

6

2.19

4

1.81

2

0.85

1

3.05

5

y Household

r = 0.842

R2 = 0.71

Is there a significant linear correlation?

= .05

n

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

25

30

35

40

45

50

60

70

80

90

100

.950

.878

.811

.754

.707

.666

.632

.602

.576

.553

.532

.514

.497

.482

.468

.456

.444

.396

.361

.335

.312

.294

.279

.254

.236

.220

.207

.196

.999

.959

.917

.875

.834

.798

.765

.735

.708

.684

.661

.641

.623

.606

.590

.575

.561

.505

.463

.430

.402

.378

.361

.330

.305

.286

.269

.256

Is there a significant linear correlation?

n = 8 = 0.05 H0: = 0

H1 : 0

Test statistic is r = 0.842

Critical values are r = - 0.707 and 0.707

(Table R with n = 8 and = 0.05)

TABLE R Critical Values of the Pearson Correlation Coefficient r

Is there a significant linear correlation?

0.842 > 0.707, That is the test statistic does fall within the

critical region.

Therefore, we REJECT H0: = 0 (no correlation) and conclude

there is a significant linear correlation between the weights of

discarded plastic and household size.

Reject

= 0

Fail to reject

= 0

Reject

= 0

1

- 1

r = - 0.707

0

r = 0.707

Sample data:

r = 0.842

(follows format of earlier chapters)

To determine whether there is a significant linear correlation between two variables

Two methods

Both methods let H0: =

(no significant linear correlation)

H1:

(significant linear correlation)

Formal Hypothesis TestTest statistic: r

Critical values: Refer to Table R

(no degrees of freedom)

Method 2: Test Statistic is r

(uses fewer calculations)

Test statistic: r

Critical values: Refer to Table A-6

(no degrees of freedom)

Method 2: Test Statistic is r

(uses fewer calculations)

Reject

= 0

Fail to reject

= 0

Reject

= 0

r = 0.811

1

r = - 0.811

0

-1

Sample data:

r = 0.828

(follows format of earlier chapters)

Test statistic:

r

t =

1 - r 2

n - 2

Critical values:

use Table T with

degrees of freedom = n - 2

significance

level

Calculate r using

Formula 9-1

r

The test statistic is

The test statistic is

r

t =

Critical values of t are from

Table A-6

1 - r 2

n -2

Critical values of t are from Table A-3 with n-2 degrees of freedom

If the absolute value of the

test statistic exceeds the

critical values, reject H0: = 0

Otherwise fail to reject H0

If H0 is rejected conclude that there

is a significant linear correlation.

If you fail to reject H0, then there is

not sufficient evidence to conclude

that there is linear correlation.

Start

Let H0: = 0

H1: 0

METHOD 1

METHOD 2

Testing for a

Linear Correlation

Why does the critical value of r increase as sample size decreases?

A correlation by chance is more likely.

Coefficient of Determination(Effect Size)

r2

The part of variance of one variable that can be explained by the variance of a related variable.

(x -x) (y -y)

r=

(x, y) centroid of sample points

(n -1) Sx Sy

x = 3

y

x - x = 7- 3 = 4

(7, 23)

•

24

20

y - y = 23 - 11 = 12

Quadrant 1

Quadrant 2

16

•

12

y = 11

(x, y)

•

8

Quadrant 3

Quadrant 4

•

•

4

x

0

0

1

2

3

4

5

6

7

Download Presentation

Connecting to Server..